I think time limits in dungeons should be more flexible and dynamic at the very least. How about just showing the time remaining until speed bonus is gone but can be extended and you can still complete the dungeon at your leisure.

Have a set timer start at 15~20m and count down to 0, by doing [Objective A] you add 10 minutes onto the timer. You get the first boss and kill him for an additional 15 minutes on the timer. [Side Objective A] would be out of your way and could make you miss the speedrun but grant you an additional 15m and an extra chest at the end.

Something like that would make the timers much more meaningful. It worked in Dyanmis (even though you still got booted out after X time, but that was so you couldn't just farm coins forever)